New INVESTIGATION OF THE STRESS-STRAIN STATE OF...

A R C H I V E O F M E C H A N I C A L E N G I N E E R I N G

VOL. LXI 2014 Number 4

10.2478/meceng-2014-0034Key words: stress-strain state, pipeline, Poisson ratio, plastic deformation, anisotropy, inner pressure, strain sensor,

stress-strain diagram

JERZY MALACHOWSKI ∗, VOLODYMYR HUTSAYLYUK ∗∗, PETR YUKHUMETS ∗∗∗,ROMAN DMITRYENKO ∗∗∗, GRIGORII BELIAIEV ∗∗∗, IHOR PRUDKII ∗∗∗

INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESSPIPE IN THE INITIAL STATE

Mechanical properties of the pipeline samples that had been cut in annular andaxial directions were investigated. The methodology of modeling and calculation ofthe real stress-strain state was described. The stable state during in the deformationprocess was defined. The results of the experimental researches were used as a testvariant during examination of pipe strength.

1. Introduction

Pipeline systems are important for effective functioning of the industryand as well as for providing the needs of the population with essential energyresources: oil, natural gas, liquid petroleum, etc. An extensive network ofpipelines, both local and internationally are supporting vital functions andeconomic development. Destruction of even small sections of the pipelinecould have cause serious consequences. Breakdown of pipelines is oftenaccompanied by explosions and fires. Total damages would result in cost-concerned with a loss of gas and petroleum products, high costs of therepair and breaking cutting the gas supply. The last one may be particularlynoticeable due to the fact that pipelines are usually located at the areasremote from settlements, which increases the time of the repair. Moreover,

∗ Military University of Technology, Faculty of Mechanical Engineering, Department ofMechanics and Applied Computer Science, Gen. S. Kaliskiego 2. St., 00-908 Warsaw, Poland;E-mail: [email protected]

∗∗ Military University of Technology, Faculty of Mechanical Engineering, Departmentof Machine Design, Gen. S. Kaliskiego 2. St., 00-908 Warsaw, Poland; E-mail: [email protected]

∗∗∗ The E.O.Paton Electric Welding Institute, Department of new structural forms of weld-ed structures and designs, No.12, Bozhenko 17.St., Kiev, Ukraine; E-mail: [email protected]

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596 J. MALACHOWSKI, V. HUTSAYLYUK, P. YUKHUMETS, R. DMITRYENKO, G. BELIAIEV, I. PRUDKII

by increasing pipe diameter and operating pressure in case of an accident itcan lead to a significant environmental pollution caused as a result from aspill of large volumes of oil or gas.

Considering life-time of the pipeline and the fact that more than 20% ofthe pipes active service life will be expired shortly, the problem of investi-gating the causes of failure of the transport systems is becoming increasinglyimportant.

A failure of the bearing elements of piping systems occurs for a varietyof reasons. It depends on the design of the elements, features of their workand loading, and physical-mechanical properties of the base material, etc.[1]. The main pipeline remains in a complex stress state under high internalpressure, temperature difference, uneven ground pressure, elastic distortionsthat result from the rugged topography, soil impacts associated with changesin its structure-swelling, drawdown, etc. These factors cause stresses frombending with stretching or bending with compression in the pipeline.

One of the main criteria that determine the reliability of pipelines arethe parameters of ultimate deformation. The change in the stress-strain state(SSS) to the level of metal plastic deformation leads to a change in geometryof the pipe and weakens the overall strength of the construction. Problemof investigating the real stress-strain state is that, a pipeline is under theinfluence of multiaxial stresses, which change during operation time. Defi-nition of all real values stresses for the mathematical design is not alwayspossible. Generally, for design of strength of the material in these cases, useda simplified scheme [2], where the stresses are measured in two planes ofmeasurement.

2. Experimental and design works

The object of investigation of this article is the seamless hot pipe pro-duced by ”INTERPIPE Nizhnedneprovsky TRUBOPRAKATNY PLANT.”Its certificate is as follows: pipe No 443, smelting number 32416, diameter219 mm, wall thickness 6 mm, weight per meter ∼33.51 kg, Steel 20. A cer-tificate of pipe quality corresponds to GOST 8732-78 and GOST 8731-74.The mechanical properties and chemical composition are shown in Table 1and Table 2, respectively.

Table 1.Mechanical properties

σ, MPa σ0.2, MPa δ,% Flattening

Melt No 32416475.78 323.73 32.0 satisfactory

480.69 328.64 33.0 satisfactory


INVESTIGATION OF THE STRESS-STRAIN STATE OF SEAMLESS PIPE IN THE INITIAL STATE 597

Table 2.Chemical composition

C, % Mn, % Si, % S, % P, % Cr, % Ni, % Cu, %Certificate(MeltNo 32416) 0.19 0.54 0.29 0.02 0.011 0.07 0.05 0.08

Measured(lab. EWI). 0.177 0.55 0.289 0.018 0.008 0.078 0.065 0.070

Mechanical tensile tests were carried out on samples in the initial state(state of delivery). At least five selected samples were cut from tube No 443in the annular and axial directions (Fig. 1).

Fig. 1. Schematic representation of the pipe

The first batch of samples consisted of four pieces cut in the circum-ferential direction. Two of them were straightened on a press, and the othertwo were not rectified with a working part. Only the place under the gripswas aligned (Fig. 2a). The second batch consisted of two samples cut in theaxial direction. The surface of the samples was not treated. The tests wereconducted at the GS Pisarenko Institute for Problems of Strength of NASU.

Fig. 2. Photo (a) and schematic illustration (b) of samples cut in the circumferential direction



The deformations in tensile specimens were defined through readingsof movements recorded by a strain sensor. The sensor was mounted on theworking parts of the samples. The strain sensor base was 25 mm. To measurethe residual deformation of the samples, the base of length of approximately40 mm was used (Fig. 3). The cross-section area of the samples was about103 mm2.

Fig. 3. Photo of the strain sensor mounted on the sample

During the mechanical test of the specimen a tension curve σ̄, ε̄ wasprepared, where σ̄ – stress calculated as the quotient of force on the areaof cross-section sample in origin state, and ε̄ = ∆l/l0 regular longitudinaldeformation of the specimen, defined as base elongation divided by the lengthof the base stamped on the working part. The maximum stress that wasrealized was called the temporary resistance. Before this value is reached,the specimen deforms uniformly, after that a neck is formed.

For application of the test results, not only in the case of a uniaxial stressstate, but also for the SSS, from standard curve of tension σ̄, ε̄ get real curveof tension σr , εr (1, 2). This assertion is valid only until the moment whenneck starts to form:

εr = ln (1 + ε̄) , σr = σ̄ (1 + ε̄) (1)

During the formation of the neck, a multiaxial stress state develops, whichdepends on the geometric shape of the specimen. The next curve is trans-formed into the so-called real strain curve – σi, εi by applying formula (3)[3]:

εi = εr − 1 − 2µ3E

σr , σi = σr (2)

where µ is the Poisson’s ratio.



In the elastic region, the Poisson’s ratio is about 0.3. Its further changeis defined by formula (3). However, the increase is observed only at theinitial stage prior to the yield strength εi0.2. It results from generation of newsources of dislocation and an increase in their density [7]. An example ofcalculation of Poisson’s ratio µ through σ̄, ε̄, is shown in Figs 4a and 4b.

µ =12− 1 − 2µe

2Eσr

εr(3)

where µe, E − coefficient and modulus of elasticity in the range.Residual deformation after the destruction of the sample was measured

on two different bases including a neck. This makes possible to determineresidual deformation for any base length. Outside the neck, which takes asignificant part of the deformation, the sample is deformed unevenly, and thestrain does not correspond to the tensile strength. As it is known, for thevalues greater than σ̄u, the force that stretches the sample begins to fall.

With the increasing base, the residual deformation will tend to becomeuniform and in infinity they will be equal mutually. Very significant defor-mation concentrates in the center of the neck, measurements in this placehave not been conducted.

The thin-walled cylindrical and spherical shells lose their ability to sus-tainable uniform deformation, where their internal pressure does not increases(dp = 0) [8]. When shell wall local thinning takes place, it results in the lossof stability of the original form and initial equilibrium of construction.

3. Results and Discussion

Figure 4a shows the tension curve obtained from the experimental dataand adapted her for real stresses and strain in axial and ring samples. In thelower part figure, there is shown the calculated change in Poisson’s ratio [3].It is worth noting that its initial value is 0.3. Curves are obtained up to startforming neck. Figure 4b present the tension curve and the calculated valueof Poisson’s ratio up to 0.05% strain.

Points on the tensile diagram correspond to the maximum value (Fig. 4).They present horizontal part on the real curves, respectively.

Elastic deformation (indicated by E in the figure) was obtained on theassumption that the modulus of elasticity E = 2·105 MPa. For circumferentialand axial specimens, the proportional limit corresponds to the value of theorder of 200 MPa.

Relations rate of strain from the level of deformation is shown in Fig. 5a.Relations of the deformation time, from the level of deformation is shownin Fig. 5b. Tensile strain rate was defined as ∆ε̄/∆t and plotted as a function



Fig. 4. Stress-strain diagrams σ̄, ε̄, actual diagram σi, εi in the ring and axial directions, µ –change of Poisson’s ratio to 22% strain (a), up to 0.05% strain (b)

Fig. 5. Strain rate dependence (a) and time plastic deformation (b) from degree of deformationε̄, εi

of ε̄. Strain rate, defined as ∆εi/∆t and built according to εi corresponds toreal values. The curves are obtained up to strain values of 22%.



Extension rate of the working parts of the samples in the elastic re-gion, before the formation of neck occurred, was less than 2·10−4, 3·10−4,7·10−4 s−1 for the circular sample and 2·10−4, 6·10−4, 4·10−4 s−1 for the axialsample. In the initial part of the plastic deformation for the axial sample, theyield plateau is observed.

Figure 6 shows the partial curves built for the circular and axial sampleswith the strain of not greater than 5%. The number of measurements per-formed by the strain sensor is about 12 measurements in the first 5 secondsof stretching and 12 measurements in the last 100 seconds. The total numberof measurements for the axial sample is 103 and 116 for the circular one.

Fig. 6. Relations of strain rate (a) and time of plastic deformation (b) from level of deformationin the interval deformation 0–5%

The graphs show that the pipe in both circular and axial directions in-dicates peculiar anisotropy. If the curves obtained for the circular directionmoves to the right direction by 0.02, they approximately will be the samewith the curves obtained from the axial direction. In the circular direction,the pipe deformed up to 2%, which is associated with its production. It isknown that after the manufacturing process, an internal pressure test was notcarried out.



Table 3 shows the mechanical properties obtained by testing tensile spec-imens.

Table 3.The mechanical properties obtained by testing tensile specimens

Ring samples Axial samples(III)

Deformationworking part (I)

Withoutdeformation

working part (II)

a b a b∗ a∗ b

σ̄y, MPa 320 320

σ̄0.2, MPa 320 327 315 308.5

σ̄u, MPa 473.21 475.5 479.34 474.76 462.39 466.67

δ, % along base of extensometer – 38.99 35.96 41.04 – 44.79

δ, % along base of sample 26.76 33.35 33.77 33.13 40.97 37.73

δ, % uniform deformation 13.9 20.3 17.2 18.6 21.2 18.8

ψ, % relative reduction 44.52 50.79 48.98 48.24 54.6 54.9∗ – samples for constructing curves. Base of stress sensor – 25 mm. Base of sample – 41 mm.

Table 3 shows that for both directions ”yield stress” σ̄0.2 and tensilestrengths σ̄u are the same. Figure 7 shows the stress-strain curve of sixsamples.

Fig. 7. The stress-strain curve of six samples. On the descending part of the curve of samples Iaand IIIa*, stress sensor was removed

Determination of the steady state during the deformation was carriedout according to the method of calculation for thin-walled cylindrical andspherical shells loaded by internal pressure. For this purpose, the curve dataσi, εi for ring samples was used.



Approximation of the actual strain curve of steel (Fig. 8) σi, εi, whereσi = σ̄ (1 + ε̄) εi = ln (1 + ε̄) , from σy, εy to start of neck formation σiu, εiucan be calculated with the use of the following equation:

σi = σy

(εi

εy

)m(4)

where σy, εy = σ0,2, ε0,2, m = 0.154253.

Fig. 8. Phase stable deformation sample made from thin-walled cylindrical and spherical shellsthat was loaded by internal pressure (a), scaled part curve (b)

Table 4.Maximal values intensity of strain and stress stable deformation

σi = 740.0249ε0.154253i σi = 87.17845 ln εi + 715.03

I II

εi σi, MPa εi σi, MPa

Stretched sample 0.1542 554.6332 0.1574 553.8401

Spherical shell 0.1028 521.0067 0.1111 523.4706

Cylindrical shell 0.0892 509.7262 0.0982 512.7107



For thin-walled, cylindrical and spherical shells loaded by internal pres-sure with the initial radius equal to r0, under large plastic deformations, theexpression for the intensity of stress σi and logarithmic strain εi are actualfor cylindrical shell:

σi =

√3

2Prs, εi =

2√3

ln(rr0

)(5)

where P − pressure; r0 and r − initiate and mean radius shell; s – originalwall thickness for spherical shell:

σi =Pr2s, εi = 2 ln

(rr0

)(6)

When the loss of stability of the initial equilibrium shape of shells occurs,values r/r0 for cylindrical and spherical shells can be obtained from thefollowing expressions. [3]:

rr0

= exp√

32

(m√3− B

) = exp(m2

)(7)

rr0

= exp[12

(23m − B

)]= exp

(m3

)(8)

where m = 0.154253.The calculations show that for spherical shells relation r/r0 is smaller

than for cylindrical ones. Next, the dependence between the pressure P andthe mean radius r for cylindrical and spherical shells is obtained [3]:– cylindrical:

P =2√3

r0s0

r2 A(B +

2√3

lnrr0

)m(9)

– spherical:

P =2Ar2

ι̂ s0

r3

(B + 2 ln

rr0

)m(10)

where s0 – original wall thickness, m = 0.154253, B – constant.It is worth pointing out that in the derivation of these calculations, rela-

tions were used obtained in the condition of incompressibility of the materialfor µ = 0.5. For cylindrical shell formula: r0s0 = rs, and for spherical shellapproximation: r2

0s0 = r2s. Thus it is possible to obtain pressure P∗, whenloss of stability of equilibrium initial form shells occurs. Assuming that B = 0the following expression is obtained for cylindrical shells:

P∗ =2√3

As0

r0 exp (m)

(m√3

)m= 504.3225

s0

r0(11)



for spherical shells:

P∗ =2As0

r0 exp (m)

(2m3

)m= 893.1103

s0

r0(12)

Hence, P∗ = 28.413 MPa and P∗ = 50.31607 MPa for cylindrical and spher-ical shells, respectively. The spherical shell can withstand the pressure about1.770911 times more than the cylindrical one. By using applying obtainedresults, it is possible to develop a practical application method of calculationof the mechanical properties of metal structures and criteria for their relia-bility. The results of experimental studies can be used as screening criteriaduring the examination of structures of cylindrical shape. After productionof containers, as well as during their operation, their load span pressure Ps,which exceeds the working pressure Pw 1.5 times. If the container is builtout of steel, where the ratio of tensile strength to yield strength ≥ 2, thepressure can be reduced to 1.5 times [6].

According to 1.9 [4], pipes of all types that work under pressure, mustwithstand a hydraulic pressure test that is calculated according to the formulagiven in [5]. Where, R – allowable stress equal to 40% of tensile strengthfor this steel, s – minimal (including minus tolerance) wall thickness of thepipe, mm and D – nominal outside diameter of the pipe, mm.Dc = D − s; – calculated diameter of the pipe, mm.

sD

=6

219= 0.027.

Taking into account the nominal outside diameter (219 mm), minimum ac-ceptable standard value of tensile strength (412 MPa) and nominal wall thick-ness (6 mm) the following values are obtained:

Pt =2sRDp

= 9.285MPa (13)

Pc =Pt

1.5= 6.19 MPa (14)

Taking into consideration the average value of tensile strength obtained inthe samples cut in the circumferential direction (475.7 MPa), the values of10.72 and 7.147 MPa, are obtained for Pt and Pc, respectively.

4. Conclusions

The article presents the stress-strain curve of the samples before the startof necking. The identical samples were cut in the circular direction. The



curves tension samples cut in the axial direction are characterized with ayield plateau. The diagrams show that the pipe in both the circumferentialand axial directions indicates peculiar anisotropy. If the curve obtained forthe circular samples moves to the right direction by 0.02, they approximatelywill be the same with the curves obtained from the axial direction. In thecircular direction, the pipe deformed up to 2%, which is associated with itsproduction.

Determination of the steady state during the deformation was carriedout according to the method of calculation for thin-walled cylindrical andspherical shells, that was loaded by internal pressure.

Hence, P∗ = 28.413 MPa and P∗ = 50.31607 MPa for cylindrical andspherical shells, respectively. The spherical shell can withstand the pressureabout 1.770911 times more than the cylindrical one.

The results of experimental studies can be used as screening criteriaduring the examination of structures of cylindrical shape.

Taking into account the nominal outside diameter (219 mm), minimumacceptable standard value of tensile strength (412 MPa) and nominal wallthickness (6 mm) the following values are obtained:

Pt = 9.285MPa; Pc = 6.19 MPa;

Taking into consideration the average value of tensile strength obtained in thesamples cut in the circumferential direction (475.7 MPa), the values 10.72and 7.147 MPa are obtained for Pt and Pc, respectively.

Acknowledgment

The authors gratefully acknowledge the funding from Seventh Frame-work Program, Marie Curie Actions: “Innovative nondestructive testing andadvanced composite repair of pipelines with volumetric surface defects”;Acronym: INNOPIPES; Proposal Number: 318874; Grant Agreement Num-ber: PIRSES-GA-2012-318874

Manuscript received by Editorial Board, May 14, 2014;final version, September 28, 2014.

REFERENCES

[1] Borodavkin P.P., Edited by Sinykov A.M.: Strength of pipelines. Moskow, Nedra, 1984.[2] Androno I.N., Kuzbogev A.S., Aginey R.V.: Resource of ground pipelines – I: Factors that

constraining resource. Standard Test Methods, Ukhta, 2008, pp. 75-96.[3] Malinin N.N.: Applied theory of plasticity and creep. Second Edition. Moskow, Mechanical

engineering, 1975.



[4] GOST 8731-74 – Steel seamless hot rolled pipes. Technical requirements.[5] GOST 3845-75 – Metal pipes. Method of hydraulic pressure test.[6] Safety regulations 03-576-03 – The rules for design and safe operation of vessels that are

working under pressure.[7] Frost H.J., Ashby M.F.: Deformation-mechanism maps. Chelyabinsk, Metallurgy, 1989.[8] Mildin A.M., Zelentsov D.G., Olevskiy V.I., Pletin V.V.: Developing the concept of predic-

tion the mechanical properties of thin-walled shells. Eastern-European Journal of EnterpriseTechnologies, 2011, Vol. 1, No. 3, pp. 57-61.

Badanie naprężeń i odkształceń rury bezszwowej w stanie wyjściowym

S t r e s z c z e n i e

W pracy przedstawiono wyniki badań właściwości mechanicznych materiału z próbek wycię-tych z rury bezszwowej na kierunku obwodowym i osiowym. Zaproponowano metodykę mode-lowania i szacowania stanu naprężenia i odkształcenia. Jest ona zgodna z metodologią stosowaną doanaliz wytrzymałościowych cienkościennych cylindrycznych i sferycznych zbiorników poddanychciśnieniu wewnętrznemu. W weryfikacji wyników wykorzystano wyniki badań doświadczalnych.


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